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Hi folks, this is Matt again. 
And, now we are ready to start solving 

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games, and making some predictions of how 
people play, in different settings. 

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And, so we are talking right now about 
Nash Equilibrium, which is one of 

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probably the most basic in standard 
solution concept of all in all the game 

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theory. 
Which is named after John Nash, who was a 

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mathematician in Princeton, and actually 
some years back won a Nobel Price, for 

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his work on this subject. 
And it's a, it's a very basic and 

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fundamental concept, and in order to sort 
of motivate it, let's, let's start by 

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just talking through some a particular 
game that was described and invented by 

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another famous person. 
so this is John, John Maynard Keyne's 

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Beauty Contest game. 
So what's the idea here? so let's, let's 

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think of a basic situation that you might 
be interested in. 

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and this was one that Keynes described in 
some detail. 

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So the idea was you have a stock, you're 
holding on to it. 

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And the stock price is rising, 
that's great. 

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you're an investor, you're trying to make 
profits off of your stock holdings. 

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And you begin to believe that maybe the 
stock is too high to be justified by the 

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value of the company. 
So, you're thinking that it's possible 

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this stock is overvalued, 
maybe there's a bubble in the market and 

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you're starting to think about selling. 
Okay, well you'd like to sell it, but 

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you'd like to wait until the price is at 
it's peak. 

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Right, so you'd want to wait until the 
price is just where it's going to hit its 

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maximum before you sell it. 
so you want to get out of the market just 

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before the other investors do. So this is 
a game where now you have to predict what 

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other people think about the stock price 
and, and what they're going to do and 

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when they want to get out, so how will 
they act. 

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How should you respond to that so, the 
basic ingredients of Nash equilibrium are 

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going to be having some prediction of 
what other players are doing, and then 

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choosing the optimal strategy in response 
to that So these are going to be two key 

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ingredients that we have. 
so there's a, a very stylized version of 

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this which is known as the, the Keyne's 
beauty contest game. 

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where did it come from? Well, actually 
Keyne's described the, there was a 

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newspaper in the in, in England that had 
a contest where players had to guess 

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Which picture of, of several women other 
readers would think was the the, the most 

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attractive one? So it wasn't to guess 
what you thought But what you thought 

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other people were thinking. 
So, Keyne's likened investing to this, 

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you, it's not only what you think of, of 
the stock, but what you think other 

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people are thinking about the stock 
that's important in driving your 

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decisions. 
Okay, 

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so, so this now is represented by a very 
simple game which is played by played by 

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many people. 
So what's this game look like? Each 

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person gets to name an integer between 1 
and 100. 

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Okay? So you get to pick a number between 
1 and 100, it has to be an integer. 

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So 1, 2, 3, etcetera. 
players are going to move simultaneously. 

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And the player who names the integer 
that's closest to 2/3 of the average 

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integer wins a prize. 
and the other players get nothing. 

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So to win this game, you have to guess, 
you have to guess the average and then 

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2/3 of it, right? So you'd want, you want 
to be right at 2/3 of whatever the 

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average guess is. 
So a little bit below the average guess. 

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if there's 2 people who happen to hit the 
same integer that, that's the right one 

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then ties are going to be broken 
uniformly at random. 

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So we'll just flip a coin or if there's 3 
people we'll, we'll roll the dice 3 sided 

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die, etcetera. 
Okay, so how would you play this game? 

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you have to think about what other 
players are going to do and then forecast 

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what the best integer is, in a response 
to that.